4. Levered and Unlevered Cost of Capital. Tax
Shield. Capital Structure
1.1 Levered and Unlevered Cost of Capital
Levered company and CAPM
The cost of equity is equal to the return expected by stockholders. The cost of equity
can be computed using the capital asset pricing model (CAPM), the arbitrage pricing theory
(APT) or some other methods.
According to the CAPM, the expected return on stock of an levered company is
(1) R E = R F + β E (R M − R F )
RE is the expected rate of return on stock of an levered company (levered cost of equity
RF is the risk-free return,
β E is the beta coefficient of stock of an levered company, and measures the volatility of the
stock’s returns relative to the market’s returns (systematic risk), it is called levered beta,
RM is the expected return on the market portfolio,
(RM - RF) is the market risk premium for bearing one unit of market risk.
Unlevered company and CAPM
Required rate of return for the stockholders RU of an unlevered company (a company
without debt) can be also expressed as a function of risk free-rate of return and market
premium risk. This time unlevered beta coefficient is used:
(2) R U = R F + β U (R M − R F )
RU is the expected rate of return on stock of an unlevered company (unlevered cost of capital),
βU is the beta coefficient for an unlevered company (unlevered equity beta, all-equity beta).
1.2 Project Beta
Managers often assume that the cash flows from the project under consideration will be
as risky as the cash flows from the existing operations of the firm (scale-enhancing projects).
This approach is strictly valid for scale-enhancing projects. When a firm takes on a project
with cash flows that do not have the same characteristics as the firm’s existing assets, it
should use the appropriate cost of equity for specific projects and not cost of equity for the
whole company. The RRR for an investment should be project specific and not company
specific. The opportunity cost for a particular project depends on the project’s risk and should
be determined by the market.
A project’s relevant risk (βproj) should be measured in terms of the relationship between
the returns generated by the project and the returns on the market portfolio. Thus a project
beta measures the sensitivity of changes in a project’s returns to changes in the market return.
A project’s beta may be estimated using historical information when it is available.
When it is not available the following procedure may be used:
1. Find a publicly traded company whose business is as similar as possible to your project.
The company thus identified is called a “pure-play company”. It is better to work with
2. Determine the equity beta, βE, for the pure-play firm’s stock.
3. Calculate the pure play’s unlevered equity beta (asset beta), βU, from its βE, by explicitly
adjusting the βE for the pure play’s financial leverage.
4. Calculate the project’s beta βproj from its unlevered equity beta βU, by explicitly adjusting
the βU for the project’s financial leverage.
5. Find the RRR for your project, using the project’s beta βproj from Step 4.
Using Damodaran’s relationship between levered beta and unlevered beta the the
following formulas are used:
(1) βU = (Step 3)
1 + comp (1 − Tcomp )
= β U + β U (1 − Tproj ) proj
(2) β proj (Step 4)
Dcomp and Ecomp represent the market values of the pure-play company’s debt and equity, and
Tcomp is the pure play’s tax rate.
Dproj and Eproj represent the market values of the company’s debt and equity used to finance
the project, and Tproj is its tax rate.
The equity risk increases as the debt - equity ratio increases.
According to CAPM, the project’s RRR with a systematic risk level proxied by βproj
(Step 5) is
(3) R Eproj = R F + β proj (R M − R F )
1.3 Capital Structure and Firm Value
The proportion of each component of capital used by a firm determines the firm’s
capital structure. The company’s capital structure is often measured by debt-equity ratio, also
called leverage ratio. A company that has no debt is called an unlevered firm; a company that
has debt in its capital structure is a levered firm. How levered should the firm be ? Is it
possible to determine optimal capital structure ?
Optimal capital structure is the debt-equity ratio, that maximizes the firm’s value.
Theoretically it is easy to establish the optimal structure. In practice this problem is difficult
1.3.1 Optimal Capital Structure Without Taxes
Modigliani and Miller (M&M) hypothesis 1. With riskless debt and the absence of
taxes, capital structure is irrelevant. The method of financing has no impact on firm’s value.
As long as return on assets is not affected, the firm’s value is independent of its capital
structure. The most important assumptions for this irrelevance result is that capital markets
have no imperfections such as taxes, brokerage fees, and bankruptcy costs.
1.3.2 Optimal Capital Structure With Taxes
Modigliani and Miller (M&M) hypothesis 2. When a company pays corporate tax, it
should be totally debt financed. This is because debt provides valuable tax shields (tax
1.3.3 Tax Shield
One-period tax shield (tax savings for a levered company) is equal to:
(3) Tax shield = T * I
T is income tax rate
I is the interest payments (I = RFD)
D is the market value of long-term debt
According to M&M the present value of future tax savings can be found with a simple
R F DT
(4) ES = ∑
t =1 (1 + R F )
It is assumed that interest rate is equal to risk free rate and that appropriate discount rate
is the required rate of return on debt is also equal to risk free rate. It is assumed further a
constant level of debt (D) and a series of equal tax savings to infinity. According to M&M
theory value of tax shields is equal to:
R F DT
(5) ES = = TD
The value of levered firm is greater than the value of an unlevered by the present value
of tax shields:
(6) D + E = A + ES
1.3.4 Financial Distress
Financial Distress and Default Considerations
As the debt-equity ratio increases, the probability that a firm will be unable to meet its
financial obligations also increases. If the credit risk is high, financial distress or default may
The costs associated with default are often referred to as bankruptcy costs. There are
two types of bankruptcy costs: direct and indirect. Direct costs are legal and accounting fees,
reorganization costs, and other administrative expenses. Indirect costs are more difficult to
measure. Examples of indirect are lost sales, lost profits, higher interest rates (cost of debt),
and in the end the inability to invest in profitable projects because external financing sources
In the case of possible default, a levered firm’s value is lower by the present value of
expected bankruptcy costs.
Value Value Present value Present value
of = of + of - of
levered firm unlevered firm tax shields bankruptcy costs
without tax bankruptcy
with bankruptcy costs shield
1.3.5 Agency Costs
Agency Costs Considerations
Managers are expected to act in the best interest of owners, but sometimes managers act
in their own interest. The conflict between these interests is called agency costs. Agency costs
arise from the conflict between the interests of the managers and the owners of a company.
Agency costs in a levered company
It is supposed that as a firm increases its debt-equity ratio, its agency costs rise and, as a
result, the value of a levered company is lower. But other viewpoints suggest that agency
costs can decrease with debt. Since it is difficult to measure agency costs, it is uncertain to
what extent they influence the market value of a levered company.
value with tax shield bankruptcy
1.4 Levered Beta and Tax Shield
Selection the relation between levered beta βE and unlevered beta βU or alternatively the
appropriate way of calculating the value of tax shield ES is essential. It is a mistake for
example to use Damodaran’s formula for levered beta and in the same time Miller and
Modigliani formula for a tax shield.
The precise relationship between levered beta βE and unlevered beta βU determines cost
of equity capital RE and in the same time uniquely determines the value of tax shield ES. The
chosen relationship or interdependent particular tax shield definition influences the value of
an asset or NPV in capital budgeting. Different tax shield definitions are presented by:
1) Modigliani i Miller1,
4) Miles and Ezzel4,
5) Harris and Pringle5,
7) Fernandez - practitioners method7.
8) Fernandez - no-cost-of-leverage,
9) Fernandez - with-cost-of-leverage,
10) Marciniak - decomposition method8.
1 F. Modigliani, M. Miller (1963), Corporate Income Taxes and the Cost of Capital: A Correction, American
Economic Review, June 1963, s. 433-443.
2 S.C. Myers, S.C., “Interactions of Corporate Financing and Investment Decisions - Implications
for Capital Budgeting”, Journal of Finance, March 1974, s. 1-25
3 M. Miller, “Debt and Taxes”, Journal of Finance, May 1977, s. 261-276.
4 J.A. Miles , J.R. Ezzell, The Weighted Average Cost of Capital”, Perfect Capital Markets
and Project Life: A Clarification, Journal of Financial and Quantitative Analysis, September 1980.
719-730. and Miles, J.A. and J.R. Ezzell, Reequationting Tax Shield Valuation: A Note, Journal of Finance Vol
XL, 5, December 1985, s. 1485-1492.
5 R.S. Harris i J.J. Pringle, Risk-Adjusted Discount Rates Extensions form the Average-
Risk Case, Journal of Financial Research, 1985, s. 237-244.
6 A. Damodaran, A (1994), Damodaran on Valuation, John Wiley and Sons, 1994, New York.
7 P.Fernandez, Valuing Companies by Cashflow Discounting: Ten Methods and Nine Theories, February 2002
The relationship between levered beta βE and unlevered beta βU and in the same time
uniquely determined tax shield ES are shown in the following table
Table 1. Levered Beta and Tax Shield
No Author Levered Beta Tax Shield
1. M&M R −g n
∑ (1 + R
R DT - R FT U R F DT
RF − g
βE = βU + βU - βD + D t
t =1 F)
RM − RF E
β E = β U + (β U − β D )
D - ES n
∑ (1 + R
R D DT
E ES = t
t =1 D)
3. Miller R DT D ES = 0
βE = βU + βU - βD +
RM − RF
4. Miles and D 1+ Ru
β E = β U + (β U - β D )1 − D
∑ (1 + R
R D DT
Ezzel 1+ R
E ES =
D 1+ RD t =1 U)
β E = β U + (β U - β D )
5. Harris and D n
∑ (1 + R
R D DT
Pringle E ES = t
t =1 U)
β E = β U + β U (1 − T )
6. Damodaran D n
R U DT - (R D - R F )D(1 - T)
E ES =
t =1 (1 + R U ) t
7. Fernandez - D n
βE = βU + βU
R D DT - (R D - R F )D
practitioners E ES =
method t =1 (1 + R U ) t
8. Fernandez -
β E = β U + [β U (1 − T ) + β D T ]
(R U T + R F - R D )D
E ES = ∑
t =1 (1 + R U ) t
9. Fernandez -
β E = β U + (β U - β D )(1 − T )
∑ (1 + R
R U DT
no-cost-of- E ES = t
leverage t =1 U)
10. Marciniak R − g Bt -1 n
∑ (1 + R
R DT - R CT U R C BT
R E − g D t -1 ES =
βE = βU + βU - βD + D t =1 E)
RM − RF E
8 Z.Marciniak, Value and Risk Management using Derivatives, Oficyna SGH. Warszawa, December 2001.
βU is unlevered beta
βE is levered beta
βD is debt beta
RU is unlevered cost of capital
RE is levered cost of equity
RF is risk-free return
RC is coupon rate
RD is interest rate (yield),
RM is market rate of return
(RM - RF) is market risk premium,
T is income tax rate
E is equity including tax shield
D is long-term debt
B is book value of debt
ES is tax shield
g is growth of tax shield.